Introduction To Modern Planar Transmission Lines. Anand K. Verma

Introduction To Modern Planar Transmission Lines - Anand K. Verma


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alt="images"/>. It is shown in the first quadrant of Fig (5.7). The phase of the propagating EM‐wave in the DPS medium lags while traveling in the images‐direction.

      In the case of a DNG medium, both permittivity and permeability are negative. Using Maxwell's equations (5.5.2c,d), and replacing ∇ → − jk the wavevector triplet‐ images relations for the DNG medium are written as follows:

      (5.5.3)equation

      However, in the above expression reversal of the direction of the magnetic field involves the reversal of the direction of the power flow toward the source. Physically, it is not possible, so the above equations are rearranged as follows by associating the negative sign with the wavevector images:

      (5.5.4)equation

      The above wavevector triplet‐ images is shown in Fig (5.8b), i.e. in the left‐hand (LH) coordinate system, so the DNG medium is also called the left‐handed, LH‐medium. Both the power‐vector and wavevector triplets and their combination are further shown in Fig (5.8d). The field vectors are rotated to maintain the power flow in a positive direction. Figure (5.8d) shows that the phase and group velocities are opposite to each other (images) as for a DNG medium the vectors images and images are antiparallel, i.e. images. The DNG occupies the third quadrant in the (μr, εr)‐plane as shown in Fig (5.7).

      In conclusion, a DNG medium supports the backward wave propagation, whereas a forward wave is supported by the DPS medium. As the phase velocity travels toward the source, while energy is traveling from the source to a load, a propagating EM‐wave in a DNG medium, in the direction of the vector images, has a leading phase. This is a unique property of the DNG medium. It significantly influences the EM‐wave characteristics of the DNG medium [J.8, B.6, B.10].

      Refractive Index of DNG Medium

      The above discussion shows that Maxwell's equations in the DNG medium are written in the LH‐coordinate system. However, the wave equation (4.5.32) of chapter 4 for the DPS medium remains valid for a lossless (σ = 0) DNG medium. It provides the following expressions for the propagation constant β = kDPS and refraction index of a DPS medium:

      (5.5.5)equation

      The evaluation of the square root of negative permeability and negative permittivity is a critical issue in the DNG medium. The negative number (−1) is exp(±jπ). However, to meet the physical condition, discussed in subsection (5.5.3), we take {−1 = exp(−jπ)} [J.8, J.9]. Therefore, the square roots of negative permeability and negative permittivity are obtained as follows:

equation

      It is interesting to note that the refractive index for a DPS medium nDPS is a positive quantity, whereas for a DNG medium nDNG is a negative quantity. So the metamaterials are also known as the negative refractive index materials, i.e. the NIM. Snell's law of refraction for a DNG medium is also modified accordingly. The negative refractive index also shows the reversal of the direction of the phase velocity of the EM‐wave. However, first let us discuss the intrinsic impedance, i.e. the wave impedance for the DNG and SNG media.

      Wave Impedance of DNG and SNG Media

      Following equation (4.5.26b) of chapter 4, the wave impedance ηDNG in a DNG medium is written below:

      (5.5.7)equation

      Like the wave impedance in a DPS medium ηDPS, the wave impedance in the DNG medium ηDNG is a positive quantity; showing the outward power flow from the source into a DNG medium. However, the wave impedances of the ENG and MNG media are reactive due to nonpropagating evanescent mode:

      (5.5.8)equation

      The inductive/capacitive reactive wave impedances of the ENG and MNG media create the reflecting surfaces. The circuit model of the metamaterials, discussed in section (5.5.3), elaborates on the nature of the RIS. Further details of the artificial RIS surface is discussed in section (20.2) of chapter 20. It is noted that the ENG/MNG medium is realized through the nonpropagating evanescent wave. Such an environment is provided by a rectangular waveguide below the cut‐off region. It is commented in subsection (7.4.1) of chapter 7.

      The propagation constants of EM‐waves in the ENG and MNG media are obtained below:


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